Koch, H. Algebraic number fields. (English) Zbl 0819.11044 Parshin, A. N. (ed.) et al., Number theory II: Algebraic number theory. Transl. from the Russian. Berlin: Springer-Verlag. Encycl. Math. Sci. 62, 1-269 (1992). This survey of classical algebraic number theory, including class field theory, Galois theory of local and global fields, Iwasawa’s theory, \(p\)- adic \(L\)-functions and Fröhlich’s theory is a translation of the Russian edition [Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 62 (1990; Zbl 0722.11001)] with removed innumerous printing errors of the original.For the entire collection see [Zbl 0814.00007]. Reviewer: W.Narkiewicz (Wrocław) Cited in 11 Documents MSC: 11Rxx Algebraic number theory: global fields 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11R23 Iwasawa theory 11R37 Class field theory 11R52 Quaternion and other division algebras: arithmetic, zeta functions 11R32 Galois theory 11R54 Other algebras and orders, and their zeta and \(L\)-functions 11R47 Other analytic theory 11R29 Class numbers, class groups, discriminants 11S31 Class field theory; \(p\)-adic formal groups 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 20E18 Limits, profinite groups 11S20 Galois theory Keywords:survey; algebraic number theory; algebraic number fields; orders; Dedekind rings; valuations; Hecke \(L\)-functions; class field theory; explicit reciprocity laws; structure of Galois groups; class number; Iwasawa’s theory; \(p\)-adic \(L\)-functions; Artin \(L\)-functions Citations:Zbl 0722.11001 PDFBibTeX XMLCite \textit{H. Koch}, Encycl. Math. Sci. 62, 1--269 (1992; Zbl 0819.11044)